4.1 Hypotheses on a
We discuss two types of hypotheses, the hypothesis of no levels feed-back and the
hypothesis of a unit vector in a.
First, let xt = (x'1t, x'2t)' be a decomposition of the variables into two sets of p — m
and m variables, and decompose a = (ɑɪ,o^/ similarly. The hypothesis on no levels
feed-back
a = ( a^ = ( γ y (14)
or a2 = 0, means that the acceleration ∆2x2t does not react to a disequilibrium error
in the polynomial cointegration relations β'xt-1 + δ'∆xt 1. Expressed differently this
means that the error term ε2t cumulates to common trends and in this sense the
variables in x2t are pushing variables with long-run impact. The hypothesis of weak
exogeneity of x2t is a restriction on the rows of (α,α±1), and that is not tested here,
see however, Paruolo and Rahbek (1999).
Second, the hypothesis that a unit vector, e1, is in a, as formulated by
a = (eι,eι±φ). (15)
An equivalent way of saying this is that the first row of aɪ is zero, e'1 aɪ = 0, so that
aɪ = e1±ψ.
This has the interpretation that the errors of the first equation are not cumulating and
in this sense the variable is purely adjusting Juselius (2006 p. 200).
Both hypotheses are restrictions on the coefficient of the stationary polynomial
cointegration relations, β'xt-ι + δ'∆xt 1, and therefore the likelihood ratio tests sta-
tistics are asymptotically χ2 with degrees of freedom mr and p + r — 1 respectively,
corresponding to the number of restricted parameters.
4.2 Test on aɪi and ɑɪ2
When testing hypotheses on the adjustment coefficients a11 and a 2 it is useful to have
expressions for the asymptotic variances, so that t-test and Wald test become feasible
without having to estimate the model with the restrictions imposed on aɪɪ and a±2.
The maximum likelihood procedure determines the superconsistent estimators for the
parameters τ, p, β'τɪ, and β = τp, which can therefore be treated as known when
discussing inference on aɪɪ and a±2.
It turns out that ai1 and a 2 are functions of a and the coefficient matrix ζ =
ax T + ω' to τ'∆Xt-1. The parameters a and ζ are determined by regression of ∆2xt on
the stationary processes β'xt-1 + β'τ±τβ∆xt~1 and τ'∆xt~1. The asymptotic variance
of (a, ζ) is therefore given by
τ r / ^ 5=∖ -r _ z-κ
(16)
asVar(a, ζ) = Φ ® Ω,